Force of a set
by Marcin Kowalczyk and Marcin Sawicki
This is an abridged version of the paper submitted to the VII European Union Contest for Young Scientists in Newcastle, where it was awarded a third prize.
Many readers of Delta may have been amazed by the fact that a
segment and a square are equipotent sets, i.e., that there exists
a bijection between them (Fig. 1).
No wonder, since the discoverer of this fact himself wrote to Dedekind in a letter: "I see it, but I don't believe it". Both sets differ in dimension, but, on the other hand, dimension identifies e.g. an open segment with a closed one, whereas the latter has two more points than the first, doesn't it?
In this paper we introduce the notion of the force of a set. This is to be a characteristic feature of some sets that takes into account the intuitions suggested above and is in fact a combination of dimension and Euler's characteristic. The force of a set will always be a polynomial in one variable r.
Let's assume first that the force of the set of all real numbers
is r. Intuition suggests again that the force of an open
or closed segment, of a square or a cube should be computed as
shown on figure 2. Further suggestions related to more complex
sets are shown on Fig. 3.
Let's make our considerations precise. A finite CW-complex is,
intuitively speaking, a space which is the union of a finite family
of disjoint, "well-behaved" cells (i.e. sets homeomorphic
with cube of different dimensions). In particular, a finite CW-complex
is compact and the border of each cell is the union of a subset
of cells of lower dimension. Figure 4 should give you an idea
of what it's all about. More pedantic readers may find the exact
definition in algebraic topology textbooks (e.g. in ).
Define a function f on the cells of a CW-complex with values in a set Y (with addition and multiplication) and such that , where for each m greater or equal to 1, is an open m-dimensional cell and is a point. Now we can define the force F in the following way:
The sum extends over all the cells of the complex K. It is easy to prove that for disjoint A and B and .
The definition is quite general; by appropriate choices of Y
and f we get many different notions. For instance, if Y
is the set of all integers, ,
then this force function is well known indeed: this is Euler's
characteristic, usually denoted by .
Figure 5 features some examples of its computation.
If the above considerations are not restricted to CW-complexes, we can similarly define other notions, like the measure of a set, probability or cardinality. Such definitions may not be very practical, but they show the generality of the introduced notion.
One might presume that in order to make the definitions of force
suggested by Figures 2 and 3 precise, Y must the ring of
polynomials in the variable r and f should be the
function defined by .
However, Fig. 6 shows that this definition does not determine
the force uniquely. So, how can it be modified?
Note that in the above examples all the forces of a given set are of the same degree and, moreover, they are congruent modulo (r+1). This observation makes the solution possible. Let be a set of polynomials in one variable r with non-negative integer coefficients. The set Y, appropriate for our purposes, is obtained by identifying any two polynomials P and Q of equal degree whenever their difference is divisible by (r+1), which will be denoted by . The elements of the set Y (for the initiated: equivalence classes of the equivalence relation ) can be multiplied and added just like ordinary polynomials.
Lemma. Any two different polynomials of equal degree in the set
take different values at r = -1. Moreover, for every and each integer m there exists a polynomial Q of degree n in the set such that Q(-1)=m.
The proof consists in the consideration of some simple cases and is therefore omitted.
Theorem. For each polynomial there is exactly one polynomial Q in such that .
Proof. The claim of the theorem is obvious, when the polynomial P is a constant. If the degree of P is positive, then choose a polynomial Q of the same degree in the set such that Q(-1)=P(-1). By the lemma, the polynomial Q is uniquely determined, and Bezout's theorem implies that the difference of the polynomials P and Q is divisible by (r+1). Hence, .
We need not change the definition of the function f; it is still assumed to satisfy . Now we can prove the following:
Theorem. The force of a finite CW-complex is well defined.
Proof. We will prove that the force is well defined for a space with well defined Euler's characteristic and dimension (i.e. the largest cell dimension). Take a polynomial F(A) computed for a given partition of a given set A and the Euler characteristic of A:
As can be seen, (see also Fig. 6). But Euler's characteristic of A is well defined so for any two partitions of the set A the corresponding polynomials F1 and F2 take the same value at r = -1, i.e. they are congruent modulo (r+1). Moreover, their degrees are both equal to the (well defined!) dimension of A. This implies and consequently F1 and F2 represent the same element of the set Y.
Corollary. If the dimension and the Euler characteristic of a set A are known, then its force F(A) can be determined uniquely.
Note: Euler's characteristic is a homotopy invariant, which is not the case with the dimension. Therefore the force of a set is not homotopically invariant either.
When two sets have identical CW-complex partitions, then their forces are clearly equal. The notion of force is so apt and natural, that the inverse theorem holds true as well.
Theorem. For any two sets A and B with equal forces there exist identical partitions A1,...,An of the set A and B1,...,Bn of the set B ( "identical" means that the two cell systems are equal).
Figure 7 shows how the theorem works. We invite our readers to provide a proof (which may be somewhat technical) by themselves. The following lemma may be a useful suggestion.
Lemma. For any the cube can be decomposed into three disjoint sets: the cube , the cube and the cube (see Fig. 8).
And one more thing: the set on the left of Fig. 7 is not a CW-complex. In fact, Euler's characteristic and the force of a set seem to work in some other cases too, e.g. for partitions which do not consist of CW-complexes (see Fig. 4) or for sets that are not compact. The authors ignore how the class of CW-complexes can be extended to cover such cases, too.
 E. H. Spanier, Algebraic topology, McGraw-Hill, New York 1966.
 A. T. Fomenko, D. B. Fuchs, V. L. Gutenmacher, Homotopic topology, Budapest 1986.
 K. Kuratowski, Introduction to set theory and topology (Polish), PWN 1965.
 R. Duda, Introduction to topology (Polish), PWN 1992.